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Perspective cuts for a class of convex 0–1 mixed integer programs

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Abstract

We show that the convex envelope of the objective function of Mixed-Integer Programming problems with a specific structure is the perspective function of the continuous part of the objective function. Using a characterization of the subdifferential of the perspective function, we derive “perspective cuts”, a family of valid inequalities for the problem. Perspective cuts can be shown to belong to the general family of disjunctive cuts, but they do not require the solution of a potentially costly nonlinear programming problem to be separated. Using perspective cuts substantially improves the performance of Branch & Cut approaches for at least two models that, either “naturally” or after a proper reformulation, have the required structure: the Unit Commitment problem in electrical power production and the Mean-Variance problem in portfolio optimization.

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Authors and Affiliations

  1. Department of Computer Science, University of Pisa, Largo B. Pontecorvo 3, 56124, Pisa, Italy

    A. Frangioni

  2. Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”, C.N.R., Viale Manzoni 30, 00185, Rome, Italy

    C. Gentile

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  1. A. Frangioni
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  2. C. Gentile
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Correspondence to A. Frangioni.

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Frangioni, A., Gentile, C. Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106, 225–236 (2006). https://doi.org/10.1007/s10107-005-0594-3

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  • DOI: https://doi.org/10.1007/s10107-005-0594-3

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